This thesis is divided into three parts.
In the first part, motivated by Stackelberg differential games, we consider a ``nonclassical" control system where the dynamics depends also on the spatial gradient of the feedback control function.
Given a probability measure on the set of initial states, we seek feedback controls which minimize the expected value of a cost function. A relaxed system is considered, and compared with the "nonclassical" one. Necessary conditions for optimality and the continuous dependence of expected minimum cost are discussed, for both systems.
The second part is concerned with Stackelberg solutions of feedback type for a differential game with random initial data. The existence of a Stackelberg equilibrium solution is proved, provided that the control is restricted in a finite dimensional space
of admissible functions.
An example shows that, for a wide class of systems, where the minimal cost for the leading player would correspond to an impulsive control function, and thus cannot be exactly attained.
In the last part of the thesis we consider a continuum model of the limit order book in a stock market, regarded as a noncooperative game for n players. Motivated by the necessary conditions for a Nash equilibrium,
we introduce a two-point boundary value problem for a system of discontinuous ODEs, and prove that this
problem always has a unique solution,
Under some additional assumptions we then prove that this solution actually yields a Nash equilibrium.